Maximal inequalities in quantum probability spaces
aa r X i v : . [ m a t h . F A ] A p r MAXIMAL INEQUALITIES IN QUANTUM PROBABILITYSPACES
GH. SADEGHI , , M. S. MOSLEHIAN , and A. TALEBI , Abstract.
In the quantum setting, there are several concepts of independencebut they are insufficient for obtaining some noncommutative counterparts ofmaximal inequalities, and so we deal with notion of noncommutative indepen-dence. We then employ a new approach to establish some maximal inequalitiessuch as the strong and weak symmetrization, L´evy, and Ottaviani inequalitiesin realm of the quantum probability spaces. As an application, we give a re-lation between the sure convergence of series of jointly independent randomvariables and the convergence in measure. Introduction and preliminaries
Some of the important kinds of probability inequalities such as Kolmogorov,L´evy, and Ottaviani inequalities relate tail probabilities for the maximal partialsum of independent random variables; see [12, 17, 20]. By centering sums of in-dependent random variables at corresponding medians, Paul L´evy [11] obtainedsome maximal inequalities, which can play a similar role as Kolmogorov’s in-equalities. In fact, he proved the probability of a maximal partial sum of randomvariables exceeds some given number is concerned with the probability that thelast partial sum does so.L´evy inequalities (see also [19]) assert that if X , X , . . . , X n are independentrandom variables with partial sums S k , k = 1 , , . . . , n , then for any λ > P (cid:18) max ≤ k ≤ n ( S k − med( S k − S n ) ) > λ (cid:19) ≤ P ( S n > λ ) , and P (cid:18) max ≤ k ≤ n | S k − med( S k − S n ) | > λ (cid:19) ≤ P ( | S n | > λ ) . Many classical inequalities have been extended to the quantum setting. We referto [1, 21, 22] and the references therein for more information. In this paper, wegeneralize several classical maximal inequalities such as strong symmetrization
Mathematics Subject Classification.
Primary 46L53; Secondary 46L10, 47A30, 60F99.
Key words and phrases.
Noncommutative L´evy inequality; quantum probability space; trace;joint independence; symmetrization; maximal inequality. and L´evy inequalities in the quantum framework. The key ingredient in ourproofs is applying Cuculescu type projections to present a quantum version ofthe maximal partial sum of random variables.In what follows, we give some necessary preliminaries on quantum probabilityspaces. Throughout this paper, we denote by M a von Neumann algebra on aHilbert space H with the unit element equipped with a normal faithful tracialstate τ . The elements of M are called (noncommutative) random variables. Wedenote by ≤ the usual order on the self-adjoint part M sa of M .A closed densely defined linear operator x with domain D(x) is said to be affiliated with M if and only if u ∗ xu = x for all unitary u belonging to thecommutant M ′ of M . If x is affiliated with M , then x is said to be τ -measurable if for each ǫ > e ∈ M such that e ( M ) ⊆ D ( x ) and τ ( e ⊥ ) < ǫ , where e ⊥ = 1 − e is the orthogonal complement of e . The set of all τ -measurable operators will be denoted by L ( M ), and L p ( M ) may be regardedas the subspace of the p -integrable operators in L ( M ). It is known that theset L ( M ) is a topological ∗ -algebra with sum and product being the respectiveclosure of the algebraic sum and product. A sequence ( x n ) in L ( M ) convergesto x in measure whenever τ (cid:0) [ ǫ, ∞ ) ( | x n − x | ) (cid:1) converges to zero for any ǫ > x ∈ L ( M ), there exists a unique spectral mea-sure E from the Borel σ -algebra B ( R ) of R into the set of all orthogonal projec-tions such that for every Borel function f : σ ( x ) → C the operator f ( x ) is definedby f ( x ) = R f ( λ ) dE ( λ ), in particular, B ( x ) = R B dE ( λ ) = E ( B ). Further, if x ∈ L ( M ) is self-adjoint and t >
0, then we have the inequality τ ( [ t, ∞ ) ( x )) ≤ t − p τ ( | x | p ) , (1.1)which is known as the Chebyshev inequality in the literature (see [14]).Let P be the lattice of projections of M . Given a family of projections( p λ ) λ ∈ Λ ⊆ P , we denote by ∨ λ ∈ Λ p λ (resp., ∧ λ ∈ Λ p λ ) the projection from H ontothe closed subspace generated by ∪ λ ∈ Λ p λ ( H ) (resp., onto the subspace ∩ λ ∈ Λ p ( H )).Consequently, ( ∨ λ ∈ Λ p λ ) ⊥ = ∧ λ ∈ Λ p ⊥ λ . Two projections p and q are said to be equiv-alent if there exists a partial isometry u ∈ M such that u ∗ u = p and uu ∗ = q . Inthis case, we write p ∼ q . If p is equivalent to a subprojection q ≤ q , we write p q . We need the following elementary properties of projections. Lemma 1.1 ([16]) . Let p and q be two projections of M ; then(i) if p and q are equivalent projections in M , then τ ( p ) = τ ( q ) . AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 3 (ii) if p ∧ q = 0 , then p q ⊥ .(iii) if ( p λ ) λ ∈ Λ is a family of projections in M , then τ ( ∨ λ ∈ Λ p λ ) ≤ P λ ∈ Λ τ ( p λ ) .(iv) if p and q commute, then p ∧ q = pq . Some tools in the study of classical results still work in the extension to non-commutative setup. However, new techniques must be invented to find quantumversions of classical inequalities involving maximum of random variables. Indeed,the maximum of two self-adjoint operators does not exist, in general. To do awaywith this problem, we employ Cuculescu’s approach (see [5, 9]) to establish somenoncommutative maximal inequalities.In this paper, we intend to prove analogues of several classical maximal inequal-ities such as strong symmetrization, L´evy, and Ottaviani inequalities in the non-commutative setup. It is noteworthy that, in quantum probability theory, variousconcepts of independence have been introduced such as tensor independence, freeindependence (freeness), Boolean independence, and so on. We emphasize thatin the noncommutative case, the mentioned concepts of independence is insuf-ficient for our purpose, and so we need another relation of independence, theso-called joint independence, to obtain noncommutative counterpart of maximalinequalities. As an application, among other things, we give a relation betweenthe sure convergence of a series of jointly independent random variables and theconvergence in measure.2.
Noncommutative L´evy inequality
We start our work with the following definition.
Definition 2.1.
For a self-adjoint element x ∈ M , we say a real number m isthe median of x if the following inequalities hold: τ (cid:0) ( −∞ ,m ] ( x ) (cid:1) ≥
12 and τ (cid:0) [ m, ∞ ) ( x ) (cid:1) ≥ . The median of x is denoted by med( x ). The median of any self-adjoint element x always exists. In fact, the real number m := sup { α : τ (cid:0) [ α, ∞ ] ( x ) (cid:1) ≥ } is amedian of x . To observe this, let Σ := { α : τ (cid:0) [ α, ∞ ] ( x ) (cid:1) ≥ } , which is nonempty,since −k x k ∈ Σ. Consider a decreasing sequence ( α n ) of real numbers convergingto m . Then (cid:0) [ α n , ∞ ] ( x ) (cid:1) is an increasing sequence of projections, which stronglyconverges to [ m, ∞ ) ( x ). Therefore τ (cid:0) [ m, ∞ ] ( x ) (cid:1) = lim α n → m + τ (cid:0) [ α n , ∞ ] ( x ) (cid:1) ≤ GH. SADEGHI, M.S. MOSLEHIAN, A. TALEBI and hence τ (cid:0) ( −∞ ,m ] ( x ) (cid:1) ≥ . On the other hand, there exists a decreasingsequence ( β n ) in Σ such that β n ր m . Thus τ (cid:0) [ m, ∞ ) ( x ) (cid:1) ≥ τ (cid:0) ( m, ∞ ) ( x ) (cid:1) = τ ∞ ^ n =1 [ β n , ∞ ) ( x ) ! = lim β n → m − τ (cid:0) [ β n , ∞ ] ( x ) (cid:1) ≥ . Some properties of the median are presented in the next proposition.
Proposition 2.2.
Let x ∈ M sa , p ≥ , and α be a positive real number, then (i) if τ (cid:0) [ α, ∞ ) ( x ) (cid:1) < , then | med( x ) | ≤ α . (ii) med( x ) ≤ p k x k p . (iii) | med( x ) − τ ( x ) | ≤ p k x − τ ( x ) k p . In particular, | med( x ) − τ ( x ) | ≤ var( x ) √ , where var( x ) = τ ( x ) − τ ( x ) .Proof. (i) From [ α, ∞ ) ( | x | ) = [ α, ∞ ) ( x ) + [ α, ∞ ) ( − x ) and the assumption wehave τ ( [ α, ∞ ) ( x )) < and τ ( [ α, ∞ ) ( − x )) < . If med( x ) > α , then we get τ ( [ m, ∞ ) ( x )) ≤ τ ( [ α, ∞ ) ( x )) < , which is impossible; hence med( x ) ≤ α . Simi-larly, med( x ) ≥ − α . (ii) The conclusion can be deduced from (i) and the Chebyshev inequality (1.1) τ (cid:16) [2 p k x k p , ∞ ) ( | x | ) (cid:17) ≤ k x k pp (2 p k x k p ) p = 12 . (iii) It is enough to use (ii) with x − τ ( x ) instead of x . (cid:3) Recall that two classical random variables X and Y are independent if forarbitrary Borel measurable sets A and B , we have P ( { X ∈ A } ∩ { Y ∈ B } ) = P ( X ∈ A ) P ( Y ∈ B ). In quantum probability theory, there is no single notionof independence; cf. [7, 15]. Among noncommutative meanings of independence,freeness [23], which was introduced by Voiculescu and led to the development offree probability theory; tensor independence [6, 10] which is a straightforwardgeneralization of the notion in classical probability theory and gives us a way tocompute mixed moments from the moments of the summands; Boolean indepen-dence [13], which is related to full free product of algebras [4]; and monotoneindependence [15], which have been considered as the most fundamental one.Franz [8] studied some relations between freeness, monotone independence andboolean independence via B ˙ozejko and Speichers two-state free products [2]. AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 5
As an extension of the classical independent random variables, we say that asequence ( x k ) nk =1 in M is jointly independent if τ ( p ∧ q ) = τ ( p ) τ ( q ) , for all projections p ∈ W ∗ ( x , . . . , x j − ) and q ∈ W ∗ ( x j , . . . , x n ) and for each j > A ⊂ M , W ∗ ( A ) denotes the von Neumann algebragenerated by A .In the case of commutativity, the joint independence, evidently is equivalentto the tensor independence. Recall that a sequence ( x k ) nk =1 in M is tensor inde-pendent [6, Definition 2.5 and Remarks after that] if τ m Y i =1 n Y k =1 a ki !! = n Y k =1 τ m Y i =1 a ki ! , whenever a ki ∈ W ∗ ( x k ) (1 ≤ i ≤ m ; 1 ≤ k ≤ n ; m ∈ N ).The reader, however, should be noticed that the notion of joint independenceis different from that of tensor independence. For instance, take the followingtwo projections in N = M ( C ) with the normalized trace τ := tr: P = 13 − − − − − − and Q = 13 − − − − − − . Then τ ( P ) = and τ ( Q ) = . Note that P and Q are projections onto K = { ( x , x , x , x , ,
0) : x + x + x = 0 } and K = { (0 , x , x , x , ,
0) : x + x + x =0 } , respectively. Thus, K ∩ K = { (0 , x , x , , ,
0) : x + x = 0 } and so P ∧ Q = 12 − − and P Q = 19 − − − − − − − − . Therefore, τ ( P ∧ Q ) = τ ( P ) τ ( Q ) = < = τ ( P Q ), and so elements P and Q are jointly independent, but not tensor independent.For any self-adjoint element x ∈ M , one may construct a jointly independentoperator x ′ having the same distribution as x . Now, we consider the von Neumann GH. SADEGHI, M.S. MOSLEHIAN, A. TALEBI algebra tensor product M ⊗ M ′ in which M ′ = M . Then any member of M and M ′ can be regarded as elements of N := M ⊗ M ′ equipped with the tensor producttrace τ , which is uniquely determined by τ ( x ⊗ x ′ ) = τ ( x ) τ ( x ′ ) via the followingmaps, respectively: x ∈ M x ⊗ x ∈ M ′ ⊗ x. For any self-adjoint element x ∈ M , we put x = x ⊗ x ′ = 1 ⊗ x .Clearly, x and x ′ have identical moments in N (i.e. τ ( x k ) = τ ( x ′ k ) for every k ∈ N ), so x and x ′ have the same distributions by [6, Page 4]. Furthermore, x and x ′ are tensor independent with respect to τ by [6, Theorem 3.1] and hencejointly independent, since x and x ′ commute.In what follows, for x ∈ M we denote by x ′ jointly independent copy of x andset b x := x − x ′ . Proposition 2.3 (Strong symmetrization inequality) . Let x , x , . . . , x n be self-adjoint random variables in M . For any λ , there exists a projection p such that τ ( p ) ≤ τ n _ k =1 [ λ, ∞ ) ( b x k ) ! . (2.1) Moreover, the projection p is nonzero provided that [ λ, ∞ ) ( x k − med( x k )) is nonzerofor some ≤ k ≤ n .Proof. Put z k := x k − med( x k ) and set e := 1 , e k := k ^ j =1 ( −∞ ,λ ) ( z k ) , (2.2) p k := e k − ∧ [ λ, ∞ ) ( z k ) , (2.3)and q k := [ λ, ∞ ) ( b x k ) . Then ( p k ) k is a sequence of orthogonal projections. Infact, if 1 ≤ k < j ≤ n , then p j is a sub-projection of ( −∞ ,λ ) ( z k ) and p k is a sub-projection of [ λ, ∞ ) ( z k ), andhence p j p k = 0 for all 1 ≤ k < j ≤ n .Putting f k := ( −∞ , ( x ′ k − med( x ′ k )), we get ( λ, ∞ ) ( z k ) ∧ f k q k . (2.4)To see this, let ξ ∈ ( λ, ∞ ) ( z k ) ( H ) ∩ f k ( H ) ∩ q ⊥ k ( H ) be a unit vector; then h ( x k − med( x k )) ξ, ξ i = h z k ξ, ξ i > λ (by ξ ∈ ( λ, ∞ ) ( z k ) ( H ))and h ( x ′ k − med( x ′ k )) ξ, ξ i ≤ ξ ∈ f k ( H )) . AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 7
Therefore, it follows from med( x k ) = med( x ′ k ) that h b x k ξ, ξ i = h ( x k − med( x k )) ξ, ξ i − h ( x ′ k − med( x ′ k )) ξ, ξ i > λ. (2.5)However, ξ ∈ q ⊥ k ( H ) yields that h b x k ξ, ξ i ≤ λ , which contradicts (2.5). Now,Lemma 1.1 (ii) implies (2.4). The projections ( λ, ∞ ) ( z k ) , f k , and q k commutewith each other, because x and x ′ as well as their spectral projections commute.Therefore, p k ∧ f k ≤ ( λ, ∞ ) ( z k ) ∧ f k ≤ q k , which implies that n _ k =1 ( p k ∧ f k ) ≤ ∨ nk =1 q k . (2.6)Thus τ ( ∨ nk =1 q k ) ≥ τ n _ k =1 ( p k ∧ f k ) ! (by (2.6))= n X k =1 τ ( p k ∧ f k ) (as p k ’s are orthogonal)= n X k =1 τ ( p k ) τ ( f k ) (by the joint independence) ≥ n X k =1 τ ( p k )= 12 τ ( p ) , in which p := P nk =1 p k , and the last inequality follows from the definition ofmedian; hence we have proved (2.1).To investigate the last assertion, suppose that p = 0; so p k = 0 for all 1 ≤ k ≤ n .However, it is easy to see from (2.2), (2.3), and induction that ( λ, ∞ ) ( z k ) = 0 forall 1 ≤ k ≤ n , which is contrary to our assumptions. (cid:3) Corollary 2.4.
Let X k , k = 1 , , . . . , n , be classical random variables, and let b x k = X k − X ′ k , where X ′ k is an independent copy of X k for all k ; then for each λ ∈ R , P (cid:18) max ≤ k ≤ n ( X k − med( X k )) ≥ λ (cid:19) ≤ P (cid:18) max ≤ k ≤ n ( b x k ) ≥ λ (cid:19) . Proof.
It is easy to check that the projections p and W nk =1 q k in Proposition 2.3 cor-respond to the characteristic functions of the subsets { max nk =1 ( X k − med( X k )) >λ } and { max nk =1 ( b x k ) > λ } , respectively. (cid:3) GH. SADEGHI, M.S. MOSLEHIAN, A. TALEBI
Corollary 2.5 (Weak symmetrization inequlity) . Let x ∈ M be a self-adjointoperator, and let b x = x − x ′ , where x ′ is a jointly independent copy of x ; then,for any λ ∈ R , τ (cid:0) [ λ, ∞ ) ( x − med( x )) (cid:1) ≤ τ (cid:0) [ λ, ∞ ) ( b x ) (cid:1) τ (cid:0) [ λ, ∞ ) ( | x − med( x ) | ) (cid:1) ≤ τ (cid:0) [ λ, ∞ ) ( | x s | ) (cid:1) ≤ τ (cid:16) [ λ , ∞ ) ( | x − α | ) (cid:17) . In particular, τ (cid:0) [ λ, ∞ ) ( | x − med( x ) | ) (cid:1) ≤ τ (cid:0) [ λ, ∞ ) ( | x s | ) (cid:1) ≤ τ (cid:16) [ λ , ∞ ) ( | x − med( x ) | ) (cid:17) . Proof.
Applying the strong symmetrization inequality with n = 1 and x = x toget τ (cid:0) [ λ, ∞ ) ( x − med( x )) (cid:1) ≤ τ (cid:0) [ λ, ∞ ) ( b x ) (cid:1) . (2.7)Applying the strong symmetrization inequality with n = 1 and x = − x gives τ (cid:0) [ λ, ∞ ) ( − x + med( x )) (cid:1) ≤ τ (cid:0) [ λ, ∞ ) ( − b x ) (cid:1) . (2.8)Note that for any self-adjoint operator y it holds that [ λ, ∞ ) ( | y | ) = [ λ, ∞ ) ( y ) + [ λ, ∞ ) ( − y ). Summing (2.7) and (2.8) we get τ (cid:0) [ λ, ∞ ) ( | x − med( x ) | ) (cid:1) ≤ τ (cid:0) [ λ, ∞ ) ( | x s | ) (cid:1) . Finally, the right hand side of the second assertion can be obtained from τ (cid:0) [ λ, ∞ ) ( | x s | ) (cid:1) = τ (cid:0) [ λ, ∞ ) ( | x − α − ( x ′ − α ) | ) (cid:1) ≤ τ (cid:16) [ λ , ∞ ) ( | x − α | ) (cid:17) + τ (cid:16) [ λ , ∞ ) ( | x ′ − α | ) (cid:17) = 2 τ (cid:16) [ λ , ∞ ) ( | x − α | ) (cid:17) . Note that x and x ′ are commutating, and the above inequality easily follows fromBorel functional calculus. (cid:3) The following result can be concluded from the weak symmetrization inequality.
Corollary 2.6.
For a sequence ( x n ) in M sa and a sequence ( α n ) of real numbers,if x n − α n −→ in measure, then c x n −→ in measure and α n − med( x n ) −→ .In particular, if x n −→ in measure, then med( x n ) −→ . Corollary 2.7.
Let x ∈ M sa ; then, for any α ∈ R and p ≥ , k x − med( x ) k pp ≤ k b x k pp ≤ K p k x − α k pp (2.9) for some constant K p . AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 9
Proof. k x − x ′ k pp = k ( x − α ) − ( x ′ − α ) k pp ≤ K p ( k x − α k pp + k x ′ − α ) k pp )(by the noncommutative Clarkson inequality)= 2 K p k x − α k pp . Thanks to the weak symmetrization inequality, the left hand side follows via k x − med( x ) k pp = Z ∞ pt p − τ (cid:0) [ t, ∞ ) ( | x − med( x ) | ) (cid:1) dt ≤ Z ∞ pt p − τ (cid:0) [ t, ∞ ) ( | b x | ) (cid:1) dt. (cid:3) The next theorem provides a noncommutative analogue of the classical maximalL´evy inequality. We need the following lemma to establish our main result. Theproof is easy, and we omit it.
Lemma 2.8.
Let P and Q be two commuting projections. If ξ ∈ ran ( P ) ∩ ran (1 − P Q ) , then ξ ∈ ran ( Q ⊥ ) . Theorem 2.9 (Noncommutative L´evy inequality) . Let x , x , . . . x n be jointlyindependent self-adjoint random variables in L ( M ) with partial sums s k , k =1 , , . . . , n . Then, for each λ > , there exist two projections p and q and twosequences ( f k ) and ( g k ) of orthogonal projections such that τ ( p ) ≤ n X k =1 τ (cid:0) ( λ, ∞ ) ( f k s n f k ) (cid:1) , (2.10) τ ( p + q ) ≤ n X k =1 τ (cid:0) ( λ, ∞ ) ( f k s n f k ) (cid:1) + n X k =1 τ (cid:0) ( λ, ∞ ) ( − g k s n g k ) (cid:1)! (2.11) and ( λ, ∞ ) ( s k − med( s k − s n )) p, (2.12) ( λ, ∞ ) ( − s k + med( s k − s n )) q (2.13) for all ≤ k ≤ n .Proof. The proof is based on the sequence of Cuculescu type projections ( e k )relative to ( s k − med( s k − s n )) and the parameter λ . More precisely, consider Cuculescu type projections e k = ( −∞ ,λ ] ( e k − y k e k − ) e k − relative to ( y k ) with1 ≤ k ≤ n and e = 1, where y k := s k − med( s k − s n ). We set p k := e k − ( λ, ∞ ) ( e k − y k e k − ) = e k − ∧ ( λ, ∞ ) ( e k − y k e k − ) ; (2.14) r k := [0 , ∞ ) ( s n − y k ) = [0 , ∞ ) ( s n − s k − med( s n − s k )) ; (2.15) f k := p k ∧ r k , (2.16)for each 1 ≤ k ≤ n . Actually, we have p k = e k − − e k , and consequently if 1 ≤ k < j ≤ n , then p j p k = 0 is obvious according to the definition of the Cuculescuprojections ( e k ). Therefore, ( p k ) k is a sequence of orthogonal projections. Since f k is a subprojection of p k , so ( f k ) is a sequence of orthogonal projections, too.Then f k ≤ ( λ, ∞ ) ( f k s n f k ) . (2.17)To see this, it is enough to show that f k ( λ, ∞ ) ( f k s n f k ) or equivalently f k ( H ) ∩ ( λ, ∞ ) ( f k s n f k ) ⊥ ( H ) = { } . Let ξ ∈ f k ( H ) ∩ [0 ,λ ] ( f k s n f k ) ( H ) be a unit vector.Note that ξ ∈ p k ( H ) by (2.16). From (2.14) we conclude that ξ ∈ e k − ( H ) ∩ ( λ, ∞ ) ( e k − y k e k − ) ( H ); hence h y k ξ, ξ i = h e k − y k e k − ξ, ξ i > λ. (2.18)It follows from (2.16) that ξ ∈ r k ( H ); so h ( s n − y k ) ξ, ξ i ≥
0, whence h s n ξ, ξ i ≥h y k ξ, ξ i > λ by (2.18). Since ξ ∈ [0 ,λ ] ( f k s n f k ) ( H ), we have h s n ξ, ξ i ≤ λ , whichgives us contradiction.From (2.17), we have n X k =1 f k ≤ n X k =1 ( λ, ∞ ) ( f k s n f k ) . (2.19)Putting p := P nk =1 p k , inequality (2.10) follows observing that n X k =1 τ (cid:0) ( λ, ∞ ) ( f k s n f k ) (cid:1) ≥ n X k =1 τ ( f k ) (by (2.19))= n X k =1 τ ( p k ) τ ( r k ) (by (2.16) and joint independence) ≥ τ ( p ) (2.20)in which the last inequality can be obtained from (2.15) and the evident identity [0 , ∞ ) ( z − α ) = ( λ, ∞ ) ( z ) for any self-adjoint operator z and all α ∈ R . AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 11
By the same argument with − s k instead of s k , we get a projection q and asequence ( g k ) of orthogonal projections such that X τ (cid:0) ( λ, ∞ ) ( − g k s n g k ) (cid:1) ≥ τ ( q ) . (2.21)Now inequality (2.11) can be deduced by summing (2.20) and (2.21).Next, we show inequality (2.12). Let 1 ≤ k ≤ n , and let ξ ∈ ( λ, ∞ ) ( y k ) ( H ) ∩ p ⊥ ( H ) = ( λ, ∞ ) ( y k ) ( H ) ∩ T nk =1 p ⊥ k ( H ) be a unit vector; then ξ belongs to p ⊥ ( H ) = e ( H ). Since ξ ∈ p ⊥ ( H ), we deduce that ξ ∈ e ( H ) by applying Lemma 2.8 with u = e and v = ( λ, ∞ ) ( e y e ). Inductively, we infer that ξ ∈ e k − ( H ). Itfollows from ξ ∈ p ⊥ k ( H ) ∩ e k − ( H ) and Lemma 2.8 with u = e k − and v = ( λ, ∞ ) ( e k − y k e k − ) that ξ ∈ ( −∞ ,λ ] ( e k − y k e k − ), or equivalently, h y k ξ, ξ i ≤ λ .However, thanks to ξ ∈ ( λ, ∞ ) ( y k ) ( H ) we have h y k ξ, ξ i > λ , which yields acontradiction; so ( λ, ∞ ) ( y k )) ∧ p ⊥ = 0. Thus we reach inequality (2.12). Inequality(2.13) can be proved by the same argument. (cid:3) Now, we present the classical version of L´evy inequality.
Corollary 2.10.
Let X , X , . . . , X n be independent random variables in proba-bility space (Ω , F , P ) with partial sums S n ; then, for any λ > P (cid:18) max ≤ k ≤ n ( S k − med( S k − S n )) > λ (cid:19) ≤ P ( S n > λ ) , (2.22) and P (cid:18) max ≤ k ≤ n | S k − med( S k − S n ) | > λ (cid:19) ≤ P ( | S n | > λ ) (2.23) Proof.
The projections e k , p k in the proof of noncommutative L´evy inequalitycorrespond to the characteristic functions of the subsets E k = E k − ∩ { S k ∩ med( S k − S n ) ≤ λ } and P k = E k − ∩ { Y n > λ } with E = Ω. Note that theprojections p and q in Theorem 2.9 correspond to the characteristic functionsof the subsets { max nk =1 ( Y k ) > λ } and { max nk =1 ( − Y k ) > λ } , respectively, where Y k := S k − med( S k − S n ). Indeed, it is easy to check that n [ k =1 P k = { n max k =1 ( Y k ) > λ } . Thus, by Theorem 2.9, there exist disjoint characteristic functions F k , k = 1 , , . . . , n such that P (cid:18) max ≤ k ≤ n ( S k − med( S k − S n )) > λ (cid:19) ≤ n X k =1 P ( F k S n F k > λ )= 2 P ( ∪ nk =1 { S n F k > λ } ) ≤ P ( S n > λ ) , which ensures inequality (2.22).Iinequality (2.23) follows from { n max k =1 ( Y k ) > λ } ∪ { n max k =1 ( − Y k ) > λ } = { n max k =1 | Y k | > λ } . (cid:3) Noncommutative Ottaviani-type inequalities
The main result of this section reads as follows. As a result, we investigatethe relation between the convergence of a series of jointly independent randomvariables and the convergence in measure.
Theorem 3.1 (Noncommutative Ottaviani inequality) . Let x , x , . . . , x n be jointlyindependent self-adjoint random variables in L ( M ) with partial sums s k for k = 1 , , . . . , n . Then for each λ > there exist a projection p and a sequence ( f k ) of orthogonal projections such that min ≤ k ≤ n τ (cid:16) ( −∞ , λ ] ( | s n − s k | ) (cid:17) τ ( p ) ≤ n X k =1 τ (cid:16) ( λ , ∞ ) ( f k | s n | f k ) (cid:17) and (2 λ, ∞ ) ( | s k | ) p for all ≤ k ≤ n .Proof. The strategy is similar to that used in the noncommutative L´evy inequal-ity, so we give only a sketch of proof. Consider the sequence of Cuculescu typeprojections e k = ( −∞ ,λ ] ( e k − | s k | e k − ) e k − , ≤ k ≤ n , with respect to | s k | andthe parameter 2 λ with e = 1. We set p k := e k − (2 λ, ∞ ) ( e k − | s k | e k − )and r k := ( −∞ , λ ] ( | s n − s k | )for each 1 ≤ k ≤ n . Then ( p k ) k is a sequence of orthogonal projections. Setting f k := r k ∧ p k , we get f k ( λ , ∞ ) ( f k | s n | f k ) . (3.1)From (3.1), we have n X k =1 f k ≤ n X k =1 ( λ , ∞ ) ( f k | s n | f k ) (3.2) AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 13
Putting p := P nk =1 p k , the desired inequality follows via n X k =1 τ (cid:16) ( λ , ∞ ) ( f k | s n | f k ) (cid:17) ≥ n X k =1 τ ( f k ) (by (3.2))= n X k =1 τ ( p k ) τ ( r k ) (by the joint independence) ≥ min ≤ k ≤ n τ ( r k ) τ ( p ) . (cid:3) To present an application in convergence we need the following lemmata.
Lemma 3.2. [18, Lemma 2.1]
For any x and y in L ( M ) and t > τ (cid:0) ( t, ∞ ) ( | x + y | ) (cid:1) ≤ τ (cid:16) ( t , ∞ ) ( | x | ) (cid:17) + 2 τ (cid:16) ( t , ∞ ) ( | y | ) (cid:17) . Lemma 3.3. [3, Lemma 3 (ii)] If x ∈ L ( M ) is a positive operator, then for anycontraction a , it holds that a ∗ xa x in the sense that ( t, ∞ ) ( a ∗ xa ) ( t, ∞ ) ( x ) for any t > . The next result yields a quantum version of the convergence in measure andalmost sure convergence.
Corollary 3.4.
Suppose that ( x k ) k ≥ is a jointly independent sequence in L ( M ) with the partial sums s n . If s n −→ s in measure, then for each λ > thereexists a sequence { p n } n of projections converging to zero in L ( M ) such that ( λ, ∞ ) ( s n + k − s n ) p n for all k .Proof. Let k ∈ N and λ > s n −→ s in measure, so thereexists a positive integer n such that τ (cid:16) ( λ , ∞ ) ( | s n − s | ) (cid:17) ≤ λ λ + k ) for n ≥ n .Fix n ≥ n and consider the finite sequence { x n +1 , x n +2 , . . . , x n + k } . We have τ (cid:16) ( λ , ∞ ) ( | s n + j − s n | ) (cid:17) ≤ λλ + k for j ≤ k . In fact, it may be obtained fromLemma 3.2 that τ (cid:16) ( λ , ∞ ) ( | s n + j − s n | ) (cid:17) ≤ τ (cid:16) ( λ , ∞ ) ( | s n + j − s | ) (cid:17) + 2 τ (cid:16) ( λ , ∞ ) ( | s n − s | ) (cid:17) ≤ λ λ + 4 k + 2 λ λ + 4 k = λλ + k . Therefore, min ≤ j ≤ k τ (cid:16) ( λ , ∞ ) ( | s n + j − s n | ) (cid:17) ≥ − λλ + k > By applying the noncommutative Ottaviani-type inequality (3.1) and Lemma 3.3,we can find a projection p n,k and a sequence ( f j ) kj =1 such that τ ( p n,k ) ≤ P nj =1 τ (cid:16) ( λ , ∞ ) ( f j | s n + k − s n | f j ) (cid:17) min nj =1 τ (cid:16) ( −∞ , λ ] ( | s n + j − s n | ) (cid:17) ≤ k τ (cid:16) ( λ , ∞ ) ( | s n + k − s n | ) (cid:17) − λλ + λ ≤ k λλ + k − λλ + k = λ and ( λ, ∞ ) ( s n + k − s n ) p n,k for all k .Note that p n,k and p n,k +1 correspond to sequences { x n +1 , x n +2 , . . . , x n + k } and { x n +1 , x n +2 , . . . , x n + k , x n + k +1 } , respectively. By the construction of projectionsin the proof of Theorem 3.1 the sequence { p n,k } k is an increasing sequence; so byputting p n := W k ≥ p n,k , we have τ ( p n ) = τ _ k ≥ p n,k ! = lim k −→∞ τ ( p n,k ) ≤ λ for any n ≥ n . (cid:3) Using the same reasoning as in the proof of Theorem 3.1, we can present aquantum version of L´evy–Ottaviani inequality.
Theorem 3.5 (Noncommutative L´evy–Skorohod inequality) . Let x , x , . . . , x n be jointly independent self-adjoint random variables in L ( M ) with partial sums s k for k = 1 , , . . . , n . Then, for each λ > and for any < α < , there exist aprojection p and a sequence ( f k ) of orthogonal projections such that min ≤ k ≤ n τ (cid:0) [ − (1 − α ) λ, ∞ ) ( s n − s k ) (cid:1) τ ( p ) ≤ n X k =1 τ (cid:0) ( αλ, ∞ ) ( f k s n f k ) (cid:1) (3.3) and ( λ, ∞ ) ( s k ) p for all ≤ k ≤ n .Proof. Consider the sequence of Cuculescu type projections e k = ( −∞ ,λ ] ( e k − s k e k − ) e k − AXIMAL INEQUALITIES IN QUANTUM PROBABILITY SPACES 15 for 1 ≤ k ≤ n with e = 1. Set p k := e k − ( λ, ∞ ) ( e k − s k e k − ) ; r k := [ − (1 − α ) λ, ∞ ) ( s n − s k ) ;and f k := r k ∧ p k for each 1 ≤ k ≤ n . Note that f k ( αλ, ∞ ) ( f k s n f k ) , (3.4)and hence n X k =1 f k ≤ n X k =1 ( αλ, ∞ ) ( f k s n f k ) . Putting p := P nk =1 p k , inequality (3.3) can be concluded from n X k =1 τ (cid:0) ( αλ, ∞ ) ( f k s n f k ) (cid:1) ≥ n X k =1 τ ( f k ) (by (3.4))= n X k =1 τ ( p k ) τ ( r k ) (by the joint independence) ≥ n min k =1 τ ( r k ) τ ( p ) . (cid:3) References [1] T. N. Bekjan and Z. Chen,
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Non-commutative L p -spaces , Math. Proc. Cambridge Philos. Soc. (1975),91–102. Department of Mathematics and Computer Sciences, Hakim Sabzevari Uni-versity, P.O. Box 397, Sabzevar, Iran
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